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Mirrors > Home > ILE Home > Th. List > mptv | GIF version |
Description: Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
Ref | Expression |
---|---|
mptv | ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 3961 | . 2 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} | |
2 | vex 2663 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 301 | . . 3 ⊢ (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵)) |
4 | 3 | opabbii 3965 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} |
5 | 1, 4 | eqtr4i 2141 | 1 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∈ wcel 1465 Vcvv 2660 {copab 3958 ↦ cmpt 3959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-v 2662 df-opab 3960 df-mpt 3961 |
This theorem is referenced by: df1st2 6084 df2nd2 6085 hashennn 10494 cnmptid 12377 |
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